Formality problems in low-dimensional topology

Speaker: Associate Professor Zsuzsanna Dancso
Affiliation: University of Sydney

Abstract

The notion of formality originates in the early rational homotopy theory of the 70's, and it has become a powerful notion in many other fields including group theory and Lie theory. In knot theory formality isomorphisms provide powerful invariants, studied under the synonymous names universal quantum invariants or homomorphic expansions. The first prototype for a such an invariant is the Kontsevich integral, the result for which Kontsevich received the Fields medal in 1998.

I will describe formality isomorphisms for examples of knotted objects which form finitely presented algebraic structures. A prime example is the braid group, which has a famous finite presentation, the Artin presentation; other examples abound, from tangles viewed as a  tensor category or PROP, and even knotted graphs with their own strange and unique operations. Finding structure preserving formality isomorphisms for these classes of knotted objects leads to solving systems of equations, which turn out to be independently interesting in quantum algebra or Lie theory. This can be uses to import tools and theorems from topology to algebra and vice versa.